"Character" is the pattern of behaviors that makes up someone's personality. For example: "He does lots of drugs because he has a weak character." It can also mean a person (or person-like thing, e.g. anthropomorphic animals) in a movie, TV show, book, comic, video game, etc. For example: "Hermoine is my favorite character in the Harry Potter series." "Specialty" is something that someone is very good at doing or is very knowledgeable about, usually through lots of studying. For example: "Flatland tricks are Rodney Mullen's specialty." "I'm a physicist. My specialty is solid-state physics." "Peculiarity" in a person is a mildly strange character trait. In this context it is more or less synonymous with "eccentricity" or "quirk" For example: "Why do you always knock on my door 5 times?" "I don't know. It's just a little peculiarity." It can also mean something strange in general. For example: "A peculiarity of Stonehenge is how mathematically precise it is regarding its alignment with various celestial objects."

Dr halll's answer is excellent, but just to answer the second question: specialty isn't uncommon, and speciality is a British version so I can't say how common it is over there. Pecularity on the other hand is pretty rare. I almost never hear or read it.